Edge States in the Honeycomb Reconstruction of Two-Dimensional Silicon Nanosheets
Andrew J. Mannix1,2*, Timo Saari3*, Brian Kiraly1,2, Brandon L. Fisher1, Chia-Hsiu Hsu4, Zhi- Quan Huang4, Feng-Chuan Chuang4, Jouko Nieminen3,7, Hsin Lin5,6, Arun Bansil7†, Mark C.
Hersam2,8†, Nathan P. Guisinger1†
1Center for Nanoscale Materials, Argonne National Laboratory, 9700 South Cass Avenue, Building 440, Argonne, IL 60439, USA
2Department of Materials Science and Engineering, Northwestern University, 2220 Campus Drive, Evanston, IL 60208, USA
3Department of Physics, Tampere University of Technology, P.O. Box 692, FIN-33101 Tampere, Finland.
4Department of Physics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan
5Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542.
6Centre for Advanced 2D Materials and Graphene Research Centre, National University of Singapore, 6 Science Drive 2, Singapore 117546.
7Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA.
8Department of Chemistry, Northwestern University, 2220 Campus Drive, Evanston, IL 60208, USA
*Equal author contribution
KEYWORDS: edge states, two-dimensional materials, silicon nanosheets, silicene, surface reconstruction, scanning tunneling microscopy
ABSTRACT: Electrons confined within a two-dimensional (2D) honeycomb potential can host localized electronic states at their edges. These edge states exhibit distinctive electronic properties relative to the bulk, and may result in spin polarization or topologically protected conduction.
However, the synthesis and characterization of well-defined 2D structures which host such edge states remains challenging. Here, we confirm the presence of a two-dimensional electron gas (2DEG) and find evidence for unique edge states in the Ag-induced honeycomb surface reconstruction of silicon nanosheets (SiNS) grown on Ag(111). Atomic-scale scanning tunneling microscopy and computational modeling confirm that the electronic properties of the SiNS surface are determined by the honeycomb surface reconstruction. This surface presents ordered edge terminations with distinct spectroscopic signatures associated with the edge orientation, and calculations suggest that Rashba-type spin orbit coupling may result in spin-polarized conduction along certain edge orientations. This quantification of the electronic structure of edge states in SiNS 2DEGs will inform ongoing efforts to engineer quantum effects in silicon-based nanostructures.
MAIN TEXT
Honeycomb two-dimensional (2D) materials host exotic electronic phenomena[1,2] such as the linear electronic dispersion in graphene that results in effectively massless charge carriers.[3] In such materials, the reduced symmetry at edges is associated with unique electronic states, including emergent magnetically ordered edge states at zigzag edges in graphene[4-9] and graphitic silicon.[10-12] In the presence of strong spin-orbit coupling, this lowered dimensionality can introduce topologically protected, gapless edge modes.[13] In general, however, the experimental realization of these edge states have proven challenging; thus, these observations have generated widespread interest in artificial structures with analogous properties.[2] Material surfaces present opportunities to engineer artificial 2D materials through the introduction of foreign adatoms, which can reconstruct the surface into an intrinsically 2D, structurally heterogeneous layer with distinct electronic properties. In the limit of strong charge carrier confinement to the surface (i.e., semiconducting or insulating substrates), such reconstructed surfaces may host electronic properties analogous to conventional (i.e., atomically thin) 2D materials.
Recent studies have shown that silicon nanosheets (SiNS) grown on Ag(111)[14,15] are capped with a silver-induced, (√3×√3)R30° honeycomb chain trimer (HCT) surface reconstruction.[15-21]
Although initially identified as stacked sheets of 2D silicene,[14] subsequent characterization confirmed that the SiNS are composed of bulk-like Si(111) grown at the ultra-thin limit.[15,16] The HCT structure consists of a metallic, honeycomb network of silver trimers and interspersed silicon adatoms, which hosts a 2D electron gas (2DEG).[21,22,22,23] Although not directly derived from the structure of individual atoms, the honeycomb periodic potential may support properties similar to artificial honeycomb lattices on a 2DEG.[2,24] 2DEGs have also been observed at metal surfaces,
where lateral confinement of electrons has been demonstrated in corral[25,25] and lattice[24,26]
geometries. Additional confinement is intrinsically present in the metallic surface reconstructions of semiconductors (e.g., the HCT surface of SiNS), due to the isolation of the surface from bulk states. Previous studies of SiNS have attributed Dirac fermion properties to SiNS surfaces,[27,28]
although these observations require additional confirmation.[17,29]
In this Letter, we demonstrate that the reconstructed SiNS surface hosts a 2DEG with distinct edge states through ultra-high vacuum scanning tunneling microcopy/spectroscopy (STM/STS) and parallel first-principles calculations. These edge states are associated with unique edge terminations consisting of pseudo-armchair (pAC) and pseudo-zigzag (pZZ) edges. The edge terminations have been labeled “pseudo“ because they are defined by the hexagonally ordered
charge density of the HCT surface reconstruction and not the position of individual atoms. These results demonstrate that it is possible to realize edge states in a surface reconstruction, which suggests a new avenue towards the realization of exotic edge phenomena in condensed-matter systems.
Atomically clean, single crystal Ag(111) surfaces (99.999% purity) were prepared through repeated cycles of Ar+ ion sputtering (1 kV, ~1-5 A/cm2 ion current) and annealing (~550°C).
Silicon was deposited by electron beam evaporation (99.9999% purity) at rates of ~0.02-0.1 monolayers/min. First-principles electronic structure calculations were carried out within the framework of density functional theory using the VASP code. A tight-binding parameterization and Green's function methods were employed to construct different presentations of the LDOS and to simulate STM images and STS spectra (see supplemental information). STM images were
acquired in the analysis chamber of an Omicron Nanotechnology VT-STM system (base pressure
< 2×10-11 mBar) at 55 K, in constant current mode using electrochemically etched tungsten tips.
dI/dV mapping and STS spectra were acquired by applying a 30 mV sinusoidal AC voltage to the sample bias and extracting the dI/dV signal using a lock-in amplifier. Ag(111) surface-state reference spectra were acquired to exclude tip effects in the measured STS spectra.
Silicon deposition at substrate temperatures from 340-360°C results in the growth of SiNS with well-defined, faceted islands in the STM topography, as shown in Fig. 1(a). Simultaneously acquired dI/dV images (Fig. 1(a) inset) show significant contrast in the electronic local density of states (LDOS) between these domains and the Ag(111) substrate. Although it is possible to achieve large (~100 x 100 nm2) apparently monolayer SiNS, we generally observe thicker regions growing both above and possibly into the surface.[15] This growth regime results in islands of both monolayer step height and bilayer step height on top of the nanosheet. In this context, “monolayer”
and “bilayer” refer to multiples of a single Si(111) plane, with topographic step heights of 3.1 Å
and 6.2 Å respectively. Atomically resolved images, shown in Fig. 1(b), reveal a lobed honeycomb structure with a 6.4 ± 0.2 Å periodicity (i.e., ~3.7 ± 0.2 Å inter-lobe spacing). This observation matches the well-known Si(111)-(√3×√3):Ag HCT reconstruction (Fig. 1(c)), in which the silver trimers are imaged as collective pockets of charge density. Within terraces, we occasionally observe regions of asymmetry between the apparent honeycomb sublattices, which likely relate to distortions in the HCT structure (the inequivalent triangle or IET model) observed at low temperatures on Si(111)-(√3×√3):Ag.[19,30]
In general, the SiNS are highly facetted, with regular and clean edge morphologies indicative of well-defined edge structures. Since these edges are better defined than those of the Si(111):Ag reconstruction on bulk Si(111) wafers, their electronic structure can be quantified down to the atomic scale. A schematic illustration of unreconstructed HCT edges is shown in Fig. 1(c), revealing intrinsic periodicities of 0.64 nm along the pZZ edges and 1.1 nm along pAC edges.
Additionally, we note that the HCT honeycomb superstructure is rotated 30° from the underlying Si(111) planes, such that a pZZ HCT edge is overlaid on an armchair edge of the Si(111) plane, and vice versa. The angular relationships between pZZ and pAC edges remain the same as in graphene. Monolayer islands generally exhibit an elongated rectangular or hexagonal shape, consistent with the broken symmetry that results from the reconstruction present during growth.[31,32] The direct growth of bilayer regions results in triangular features, as in Fig. 1(a). The higher symmetry of bilayer islands suggests enhanced stability of the edge structure.[32,33] With this in mind, the monolayer-thick, hexagonal island in Fig. 1(d) can be identified with a pZZ edge termination, whereas the triangular, bilayer-thick island in Fig. 3(b) entirely exhibits pAC termination. These relationships appear to be general, although very large or spatially constrained monolayer islands will sometimes exhibit pAC edges. In this case, the pAC edges often have a greater density of adatoms and defects. Isolated bilayer-step-height islands with uniform pZZ edges have not been observed. Combining these edge-shape associations with the angular relationship between edge types and the registry of the islands with the underlying SiNS, the edge associations of entire SiNS regions can be inferred, as in Fig. 1(e). Large multilayer domains form by the coalescence of islands, primarily resulting in regions that appear as the composite of hexagonal and elongated hexagon islands. Consequently, these domains predominantly exhibit the
edge terminations of the “parent” islands. The general prevalence of pZZ-terminated, monolayer, hexagonal islands leads to fewer pAC edges of monolayer height.
The “bulk” electronic properties of SiNS are likely similar to those of the HCT reconstruction,
albeit modulated by the decreased thickness and hence increased role of the surfaces in conduction.
We adopt a theoretical model composed of a slab of five Si(111) planes, terminated by the HCT on top and H on the bottom (see supplemental Fig. S1). The electronic band structure is calculated and shown in Fig. 2(a). Experimentally, 2DEG-like quasiparticle interference (QPI) from both pZZ and pAC edges is observed, as shown in Fig. 2(b). Extracting the k-space periodicity from the scattering profile at the pZZ edge results in the dispersion shown in Fig. 2(c), where blue and red markers denote scattering from the upper and lower terrace, respectively. Evidently, the scattering periodicity on both terraces is nearly equivalent, suggesting that the monolayer difference in the SiNS thickness has minimal effect. Both SiNS surfaces differ significantly from the Ag(111) surface state (black markers). The calculated band structure agrees well with the measured dispersion, implying that the HCT model adequately captures the electronic characteristics of this surface. We note that the dispersion has apparent linearity above ~0.2 V, which contributed to early conclusions of Dirac fermion charge carriers.[27,28] However, we do not believe there to be Dirac fermions present in this case, despite the apparent honeycomb structure observed. This is perhaps due to the asymmetric basis provided by the Ag trimers within the pseudohoneycomb lattice.
Edge states were observed for both pZZ and pAC edges using STS, which reveal periodic modulations in the LDOS. Our computations, based on a material-specific effective tight-binding
model Hamiltonian informed by first-principles band structures and wave functions (see supplemental information), agree well with the corresponding experimental results, suggesting that the honeycomb reconstruction hosts edge states. Our theoretical model considers edge states arising from a number of physical origins (e.g., interruption of the lattice periodicity, change in potential at the edge, or merging of dangling bonds; in addition, the effect of Rashba-type SOC is considered). Fig. 3(a) shows a model for the pZZ edge which preserves the Ag trimers. A simulated STM topography image, shown in Fig. 3(b), reveals the asymmetry of the corrugation at the edge, where the high contrast pattern is oriented along the line between the outermost first layer silicon atom and the center of the most outward Ag trimer. Figures 3(c) and 3(d) present calculated LDOS maps oriented parallel and perpendicular to the edge, respectively. These calculated LDOS maps are plotted analogously to the experimental STM sample bias, where zero bias denotes the Fermi level. These maps distinguish between localized edge states and delocalized states (Fig. 3(c)), and show modulations in their intensity and energy along the edge (Fig. 3(d)). A comparison between Figs. 3(c) and 3(d) reveals edge states in both the filled states (approximately -1.0 V, -0.6 V, and -0.2 V) and empty states (~1.3 V). Additional data in the Supplementary Information provide more detailed analysis.
Fig. 3(e) shows a representative STM topography image of a monolayer pZZ edge, with the boxed portion rescaled to enhance contrast. The edge structure consists of continuous chevron features that accompany periodic electronic LDOS modulations in the dI/dV image (Fig. 3(f)), which are suggestive of an edge state hosted by the pseudo-honeycomb lattice of the HCT reconstruction. Both the topographic and LDOS modulations are equivalent at ~1.3 ± 0.16 nm, which is twice the structural edge periodicity of 0.64 nm. A line of STS point spectra acquired
along the edge is plotted in Fig. 3(g) (obtained along the blue line in 3(e)), showing pronounced modulation in both the filled and empty states. Perpendicular line spectra and constant-energy dI/dV images confirm that the mid-gap states are heavily localized at the edge, (see supplementary Fig. S5) and that the modulations occur at various energies, alternating between empty and filled states. In Figs. S5(C-G), the dI/dV intensity within the gap alternates between the two sub-units of the 2:1 superstructure. With these alternations superimposed, the structure would have the predicted periodicity (i.e., as in Fig. 3(b)). A period of two modulation in the edge states of MoS2
nanoislands[34,35] was attributed to edge reconstruction or electronic effects. However, similar periodic spatial inhomogeneity has previously been associated with spin-polarized edge states in the reconstructed Si(557)-Au system.[10,12] The spatial periodicity of these modulations is suppressed in monolayer pZZ edges at the Ag(111)-SiNS transition due to charge transfer from the substrate, (see supplementary Fig. S6) although the edge does appear to host diffuse states (i.e., lacking periodic modulations along the edge).
The atomic structure model and simulated STM topography of a pAC edge are given in Fig. 4(a) and 4(b), respectively. The LDOS perpendicular to the edge is plotted in Fig. 4(c), and shows edge states distributed throughout both filled and empty states. Along the edge (Fig. 4(d)), the intensity of the edge states alternates between the empty and filled state biases, which correspond to localization between the furthest-protruding Si adatoms and the outermost Ag trimers, respectively. Our observations suggest that well-defined monolayer pAC edges are rare compared to pZZ edges, possibly due to anisotropic growth kinetics. A representative STM topography image, shown in Fig. 4(e), exhibits significant electronic effects attributed to carrier scattering at the edge. Enhanced contrast at the edge, as shown in Fig. 4(f), shows a slight height difference
between the edge atoms. Similar to the pZZ edge, STS line spectra (Fig. 4(g)) show periodic modulation in the unfilled states, consistent with unreconstructed pAC edge (~1.2 nm).
Furthermore, the periodicity of the modulations as compared to topography shows that the enhanced LDOS is localized at the upward-buckled Ag clusters along the edge. High-quality pAC edges are more frequently observed in the direct growth of triangular bilayer islands, which favor pAC termination. The bilayer pAC edge shown in Fig. 4(g) demonstrates more symmetric topographical features than the monolayer and a strikingly different electronic LDOS modulation along its length, shown in Fig. 4(h). Like the monolayer pZZ edge, the modulations in the bilayer pAC edge are more evenly distributed between empty and filled states (Fig. 4(i)), in agreement with the simulated edge LDOS. Grid spectra, dI/dV maps and averaged point spectra (see supplementary Fig. S7) confirm the periodic modulation in both filled and unfilled states in accord with the calculated LDOS of Figs. 4(c,d). In particular, mid-gap states are localized at the edge. In this case, the high symmetry of the edge structure and the periodic modulations in LDOS are consistent with the simulated modulations.
These results demonstrate the emergence of 1D edge states in the 2D honeycomb surface reconstruction of SiNS. Morphologically, the SiNS present consistent structural relationships between island shape, size, thickness, and edge termination, enabling a detailed study of the electronic characteristics associated with these edges. The quality of these edges is likely due to the surfactant effect for Ag homoepitaxy, and it may be possible to utilize this effect generate such well-ordered edges directly on the reconstructed Si surface (e.g., by silicon homoepitaxy on the Ag-terminated surface). Furthermore, our general electronic characterization (i.e., Fig. 2) shows that the SiNS surface hosts a 2DEG associated with the HCT reconstruction, consistent with
previous results.[15,16] We find both pZZ and pAC edge terminations host edge states that will likely exhibit a wealth of interesting physical properties. For instance, the edge states may serve as spin filters through the spin Hall effect (see Supplemental information), which would provide control and manipulation of electron spin without incorporating typical magnetic elements, which remains an ongoing experimental challenge. These effects may be enhanced in other surface reconstructions, perhaps with heavier elements (enhanced SOC) or with a more symmetric honeycomb structure (thus providing a more symmetric basis set, and perhaps resulting in Dirac fermion charge carriers).
In the observed edges, there is a reasonable expectations that spin-polarized conduction may occur due to Rashba SOC. In graphene, only the zigzag edge hosts localized states, which are associated with the emergence of antiferromagnetic ordering in thin nanoribbons.[7] Unlike graphene, however, the HCT reconstruction exhibits pronounced structural chirality (i.e., broken mirror symmetry along the edge) in the (pseudo-) zigzag edges (see Fig. 3(a)). Evidently, this reduction in symmetry persists despite the 2:1 superstructure (c.f. asymmetric electronic structure in Figs. 3(f) and 3(g)) and likely plays a significant role in the propagation of charge and spin along the edge (see Supplemental Figs. S8 and S9). Presently, the production and characterization of atomic-scale edge states remains experimentally challenging. The emergence of edge states in a silicon surface with well-defined structure thus provides an experimentally feasible path towards the study and exploitation of such states and their associated exotic quantum phenomena.
Furthermore, our results highlight the effects achievable in dimensionally confined surface reconstructions, which may be considered a broad class of 2D materials with unique properties.
FIGURES
Figure 1. (a) Representative STM topography image of SiNS morphology showing monolayer and bilayer islands on a large SiNS terrace, with corresponding color scale. The inset dI/dV map shows electronic LDOS contrast between SiNS and Ag(111), imaged at Vsample= -0.1 V, It = 1.0 nA. (b) Atomic-resolution STM topograph of the √3 SiNS lattice (Vsample= 0.3 V, It = 1.0 nA). (c) Atomic model for the HCT structure, with edge orientations and periodicities noted. The yellow overlays schematically denote regions of enhanced charge density between the Ag trimers. (d) Atomically resolved STM topography image (Vsample = -1.0 V, It = 800 pA) demonstrating edge associations for hexagonal monolayer and triangular bilayer islands. The inset shows the derivative of topography to emphasize the lattice. (e) Large-scale STM topography image demonstrating edge associations inferred from hexagonal and triangular reference islands (Vsample
= -1.0 V, It = 400 nA).
13 Figure 2. (a) Electronic band structure of SiNS. (b) STM topography rendering of a SiNS edge with dI/dV signal colormap, showing quasiparticle interference standing waves from both edge types (Vsample = -300mV V, It = 800 pA). Inset shows FFT of this region, with a circle highlighting features associated with standing waves. (c) Dispersion extracted from standing waves on pseudo- zigzag edges (upper and lower terraces) and the Ag(111) surface, compared with the corresponding theoretically predicted dispersion.
14 Figure 3. (a) Atomic structure model for an unreconstructed pZZ edge and (b) accompanying simulated STM image (Vsample = -0.5 V). (c,d) Calculated density of states profiles (c) perpendicular and (d) parallel to the edge. (e) STM topograph of SiNS monolayer-height pZZ edge (Vsample = -0.6 V, It = 1.0 nA) with the inset showing the topography derivative. (f) dI/dV image obtained simultaneously with (e), demonstrating the local enhancement in LDOS along the edge, and the chiral character of the edge electronic structure. (g) STS dI/dV spectra obtained along the edge in (e), with periodic electronic structure in both filled and unfilled states (~1.3 ± 0.16 nm period). The scaled pZZ edge structural model is overlaid for comparison.
15 Figure 4. (a) Atomic structure model and (b) simulated STM image (Vsample = -0.5V) of a pAC edge. (c,d) Calculated density of states profile (c) perpendicular and (d) parallel to the edge. (e,f) STM topography (e) of a monolayer pAC edge and (f) the same region rescaled to emphasize the electronic effects and subtle asymmetry in edge structure (Vsample = -0.8 V, It = 2.0 nA). (g) dI/dV spectra obtained along the blue arrow parallel to the edge in (e), showing strong, periodic features in the unfilled states. The scaled pAC edge structural model is overlaid for comparison. (h) STM topography image of a bilayer step pAC edge (Vsample = -2.0 V, It = 1.0 nA). (i) dI/dV spectra obtained along the blue arrow parallel to the edge in (h), showing periodic features in the LDOS.
The scaled pAC edge structural model is overlaid for comparison.
16 ASSOCIATED CONTENT
Supporting Information. See Supplemental Material for details of calculations and additional STM/STS data.
AUTHOR INFORMATION
Corresponding Author
†Corresponding Authors: ar.bansil@neu.edu; m-hersam@northwestern.edu; nguisinger@anl.gov
Author Contributions
The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.
*These authors contributed equally.
ACKNOWLEDGMENTS
Funding: This work was performed, in part, at the Center for Nanoscale Materials, a U.S.
Department of Energy (USDOE) Office of Science User Facility under Contract No. DE-AC02- 06CH11357. A.J.M., B.K., B.L.F., M.C.H., and N.P.G acknowledge support by the USDOE SISGR (contract no. DE-FG02-09ER16109), the Office of Naval Research (grant no. N00014-14- 1-0669), and the National Science Foundation Graduate Fellowship Program (DGE-1324585 and DGE-0824162). FCC acknowledges support from the Ministry of Science and Technology of Taiwan under Grant No. MOST-104-2112-M-110-002-MY3. H.L. acknowledges the Singapore
17 National Research Foundation for the support under NRF Award No. NRF-NRFF2013-03. The work at Northeastern University was supported by the USDOE, Office of Science, Basic Energy Sciences grant number DE-FG02-07ER46352 (core research), and benefited from Northeastern University's Advanced Scientific Computation Center (ASCC), the NERSC supercomputing center through DOE grant number DE-AC02-05CH11231, and support (applications to layered materials) from the DOE EFRC: Center for the Computational Design of Functional Layered Materials (CCDM) under DE-SC0012575. This work benefited from the resources of Institute of Advanced Computing, Tampere. T.S. is grateful to Väisälä Foundation for financial support.
18 REFERENCES
[1] A. Bansil, H. Lin, T. Das, Rev Mod Phys 2016, 88, 021004.
[2] M. Polini, F. Guinea, M. Lewenstein, H. C. Manoharan, V. Pellegrini, Nature Nanotechnology 2013, 8, 625.
[3] A. K. Geim, K. S. Novoselov, Nat Mater 2007, 6, 183.
[4] Y. Kobayashi, K.-I. Fukui, T. Enoki, K. Kusakabe, Y. Kaburagi, Phys Rev B 2005, 71, 193406.
[5] K. Nakada, M. Fujita, G. Dresselhaus, M. S. Dresselhaus, Phys Rev B 1996, 54, 17954.
[6] P. Ruffieux, S. Wang, B. Yang, C. Sánchez-Sánchez, J. Liu, T. Dienel, L. Talirz, P. Shinde, C.
A. Pignedoli, D. Passerone, T. Dumslaff, X. Feng, K. Müllen, R. Fasel, Nature 2016, 531, 489.
[7] G. Z. Magda, X. Jin, I. Hagymási, P. Vancsó, Z. Osváth, P. Nemes-Incze, C. Hwang, L. P.
Biró, L. Tapasztó, Nature 2015, 514, 608.
[8] O. Yazyev, M. Katsnelson, Phys Rev Lett 2008, 100, 047209.
[9] C. Tao, L. Jiao, O. V. Yazyev, Y.-C. Chen, J. Feng, X. Zhang, R. B. Capaz, J. M. Tour, A.
Zettl, S. G. Louie, H. Dai, M. F. Crommie, Nat Phys 2011, 7, 616.
[10] S. C. Erwin, F. J. Himpsel, Nat Commun 2010, 1, 58.
[11] S. Cahangirov, M. Topsakal, E. Aktürk, H. Şahin, S. Ciraci, Phys Rev Lett 2009, 102, 236804.
[12] P. C. Snijders, P. S. Johnson, N. P. Guisinger, S. C. Erwin, F. J. Himpsel, New J Phys 2012, 14, 103004.
[13] I. K. Drozdov, A. Alexandradinata, S. Jeon, S. Nadj-Perge, H. Ji, R. J. Cava, B. Andrei Bernevig, A. Yazdani, Nat Phys 2014, 10, 664.
[14] B. Feng, Z. Ding, S. Meng, Y. Yao, X. He, P. Cheng, L. Chen, K. Wu, Nano Lett 2012, 12, 3507.
[15] A. J. Mannix, B. Kiraly, B. L. Fisher, M. C. Hersam, N. P. Guisinger, ACS Nano 2014, 8, 7538.
[16] T. Shirai, T. Shirasawa, T. Hirahara, N. Fukui, T. Takahashi, S. Hasegawa, Phys Rev B 2014, 89, 241403.
[17] S. K. Mahatha, P. Moras, P. M. Sheverdyaeva, R. Flammini, K. Horn, C. Carbone, Phys Rev B 2015, 92, 245127.
[18] J. Feng, S. R. Wagner, P. Zhang, Sci Rep 2015, 1.
[19] A. Sweetman, A. Stannard, Y. Sugimoto, M. Abe, S. Morita, P. Moriarty, Phys Rev B 2013, 87, 075310.
19 [20] K. J. Wan, X. F. Lin, J. Nogami, Phys Rev B 1992, 45, 9509.
[21] N. Sato, S. Takeda, T. Nagao, S. Hasegawa, Phys Rev B 1999, 59, 2035.
[22] M. Ono, Y. Nishigata, T. Nishio, T. Eguchi, Y. Hasegawa, Phys Rev Lett 2006, 96, 016801.
[23] T. Hirahara, I. Matsuda, M. Ueno, S. Hasegawa, Surf Sci 2004, 563, 191.
[24] K. K. Gomes, W. Mar, W. Ko, F. Guinea, H. C. Manoharan, Nature 2013, 483, 306.
[25] M. F. Crommie, C. P. Lutz, D. M. Eigler, Science 1993, 262, 218.
[26] S. Paavilainen, M. Ropo, J. Nieminen, J. Akola, E. Rasanen, Nano Lett 2016, 16, 3519.
[27] L. Chen, C.-C. Liu, B. Feng, X. He, P. Cheng, Z. Ding, S. Meng, Y. Yao, K. Wu, Phys Rev Lett 2012, 109, 056804.
[28] B. Feng, H. Li, C.-C. Liu, T.-N. Shao, P. Cheng, Y. Yao, S. Meng, L. Chen, K. Wu, ACS Nano 2013, 7, 9049.
[29] R. Arafune, C. L. Lin, R. Nagao, M. Kawai, N. Takagi, Phys Rev Lett 2013, 110, 229701.
[30] H. Aizawa, M. Tsukada, N. Sato, S. Hasegawa, Surf Sci 1999, 429, L509.
[31] K. Romanyuk, V. Cherepanov, B. Voigtländer, Phys Rev Lett 2007, 99, 126103.
[32] S. V. Khare, S. Kodambaka, D. D. Johnson, I. Petrov, J. E. Greene, Surf Sci 2003, 522, 75.
[33] B. Voigtländer, T. Weber, Phys Rev B 1996, 54, 7709.
[34] M. V. Bollinger, J. V. Lauritsen, K. W. Jacobsen, J. K. Nørskov, S. Helveg, F. Besenbacher, Phys Rev Lett 2001, 87, 196803.
[35] J. V. Lauritsen, J. Kibsgaard, S. Helveg, H. Topsøe, B. S. Clausen, E. Lægsgaard, F.
Besenbacher, Nat Nanotech 2007, 2, 53.
Supplementary information
Edge States in the Honeycomb Reconstruction of Two-Dimensional Silicon Nanosheets
Andrew J. Mannix1,2*, Timo Saari3*, Brian Kiraly1,2, Brandon L. Fisher1, Chia-Hsiu Hsu4, Zhi- Quan Huang4, Feng-Chuan Chuang4, Jouko Nieminen3,7, Hsin Lin5,6, Arun Bansil7, Mark C.
Hersam2,8†, Nathan P. Guisinger1†
1Center for Nanoscale Materials, Argonne National Laboratory, 9700 South Cass Avenue, Building 440, Argonne, IL 60439, USA
2Department of Materials Science and Engineering, Northwestern University, 2220 Campus Drive, Evanston, IL 60208, USA
3Department of Physics, Tampere University of Technology, P.O.Box 692, FIN-33101 Tampere, Finland.
4Department of Physics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan
5Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542.
6Centre for Advanced 2D Materials and Graphene Research Centre, National University of Singapore, 6 Science Drive 2, Singapore 117546.
7Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA.
8Department of Chemistry, Northwestern University, 2220 Campus Drive, Evanston, IL 60208, USA
*Equal author contribution
†Corresponding Authors: ar.bansil@neu.edu; m-hersam@northwestern.edu; nguisinger@anl.gov
Computational Methods and Detailed Results
We performed first-principles electronic structure calculations using the projector augmented-wave method [1,2] as implemented in the Vienna Ab Initio Simulation Package (VASP) [3, 4] within the framework of density functional theory (DFT). The generalized gradient approximation (GGA) [5]
is used. A vacuum layer of 20 was included in the supercell to model the HCT model of
Ag/Si(111) . Five Si(111) bilayers were used to simulate the substrate, where Si atoms in the bottom layer were passivated with H atoms (Fig. S1). The cutoff kinetic energy was set at 500 eV. A -centered 10×10×1 Monkhorst-Pack [6] grid was used for structural relaxations. Spin-orbit coupling (SOC) was included in self-consistency cycles and band computations.
In simulating dI/dV spectra and STM maps, we utilized a tight-binding parameterization of the DFT bands with 2N orbitals per primitive cell, where N is the total number of orbitals and the factor of 2 accounts for spin degrees of freedom. A spin-dependent Green’s function is used to include the effects of spin-orbit coupling [7,8]. Hopping integrals h in the Hamiltonian below are taken to be of Slater-Koster form with appropriately scaled matrix elements, and the SOC is included via matrix elements between the p-orbitals of atoms. We thus arrive at our effective Hamiltonian:
(S1)
where is the onsite energy of orbital , c† and c are the standard creation and annihilation operators, and the SOC part of the Hamiltonian [9,10] can be cast as:
(S2)
where i and j denote directions (x, y, z), pi and pj refer to p-orbitals of the same atom, and ui is a unit vector pointing in the direction of the ith orbital. Our tight-binding model correctly reproduces the DFT band structure at low energies, as shown in Figure S2.
Our STM/STS simulations of the edge-states are based on two “semi-infinite” side-cells on the two sides of the simulation cell. The semi-infinite slab is treated by recursively calculating the Green's function for the side cells, which are coupled through a self-energy term to the Green's function of the central cell, using the Dyson equation:
(S3) with
(S4)
where stands for the Hamiltonian matrix elements between the side-cell and the central cell and (S5)
The recursion is started from a non-interacting Green's function for the side cells, , with self-energy .
Our tunneling calculations are based on the Todorov-Pendry approach [11, 12] in which the differential conductance between orbitals of the tip (t) and the sample (s) is given by
(S6)
where is the electron density matrix and is the hopping integral between the tip orbital t and the sample orbital s. The implementation of the method is described in our earlier publications.
[13,14] Fig. S3 shows a collection of constant current STM maps for HCT and IET reconstructions.
The tip is assumed to be sharp and spherically symmetric in order to enhance details.
The computed STM images of Figs. 4(b) and 5(b) are obtained from a system of 30 and 20 silver atoms, respectively, with two underlying silicon bilayers being terminated by hydrogen. The edges form when the top bilayer consisting of silver and silicon atoms is partly removed. In Fig. S4, we show the structural model of the pZZ and pAC edges used in the computations and the associated STM images. For the pZZ edge, the corrugation is seen to be less steep than for the off-edge areas, while for the pAC edge, the two areas show similar corrugation.
Effects of SOC
As is seen in Fig. S8 (a), the band structure along the direction of the pZZ-edge forms a bunch of bands, some of them closing the gap between valence and conduction states. This is in contrast to the band structure of real ZZ edge of graphene and graphene analogues, where there are two non- equivalent high symmetry points K and K’ within BZ, which are connected by a topologically protected edge bands from the conduction states of K to the valence states of K’ and vice versa. [9]
Although no topologically protected edge states appear in the present pseudo-structures, the SOC part of the Hamiltonian brings in Rashba-type matrix elements to the Hamiltonian, since they effectively are proportional to a factor
.
Hence, reversing the momentum inverts the spin at those states where there is a large gradient of the potential experienced by the electrons. Obviously, the gradient of the potential at a 1D-edge of a 2D-structure on a substrate has a strong horizontal component. Hence, there should be a significant distinction between projection of the states to spin-up and spin-down polarizations. Fig. S8(b) Shows that SOC terms between the onsite p-orbitals of Ag atoms lead to significant break-up of spin degeneracy for the in-gap states and the states near the gap. The magnification of S8(b) shows that there are spin-momentum locked states right below Fermi-energy, and hence this suggests a possibility of spin-filtering for a hole doped system.
It is most interesting to consider the real space polarization density of the states near Fermi-energy, since they might be the most useful in possible control of spin currents. In Figs. S8 (c ) –( e) are shown the real space map of the different components of polarization at E=-0.07eV (1.5Å above the adsorbate layer). To emphasize the coupling of the momentum and spin polarization (as indicated in HSOC term of the Hamiltonian, Eq. (S2), only positive values of k have been taken into account.
The first impression from the figures is that the spin polarized states are, indeed, mainly localized along the 1D-edge. What is also notable, that there are non-localized states which are strongly polarized in y-direction. Evidently the spin-orientation of these non-localized states is dominated by the lack of periodicity in z-direction and momentum in x-direction, whose cross-product points to positive or negative y-direction. At the edge, the situation is quite different. The local geometry seems to have a subtle effect to the matrix elements from SOC, and hence there is a complicated pattern of the spin orientation in x- and y-directions.
In Figs. S9 (a) and (b) is shown the band structure and the corresponding spin polarization for pAC edge. In this case, the band structure bears a lot of resemblance to the band structure of silicene AC- nanoribbon in Ref. [15], especially as the valence bands are concerned. A relatively flat bands
crossing the steeper valence bands seems to appear in both the cases. The spin resolved band
structure seem to suggest that the k-dependent spin splitting would take place between the steep and flat bands. It might be significant from the point of view of applications that there are k-dependent spin decomposition to electrons with low and high effective mass. The real space projections in Figs. S9 (c) – (e) show that the spin-resolved states (chosen at E=-0.07eV, 1.5Å above the top layer) are also very strictly localized to the edge and a notably strong polarization is observed for all the directional components of the polarization-density. The y- and z-components being perpendicular to the edge direction are predominantly oriented to one direction in case of negative values of k, and hence p-AC edges might be used to filter these components of spin. However, the sign of the x- component is opposite for the two frontmost Ag atoms of the edge, which is an obvious
consequence of the open geometry of the p-AC edge.
Detailed STS Results and Discussion
Spatially resolved STS point spectra were acquired over a grid on multiple edges to obtain atomic- scale spatial mapping of the edge states, from which line spectra in Fig. 4 and Fig. 5 were extracted at the topographic edge boundary. However, the full grid datasets provide additional data that support the characterization of the edge states.
The monolayer pZZ edge is characterized in Fig. S5. Fig. S5a reproduces the topography image from Fig. 4, with the location of the grid acquisition indicated. Fig. S5b shows an STS line cut dataset extracted perpendicular to the edge, which demonstrates that the electronic states attributed to the edge are localized near the topographic boundary (yellow line). The complete grid spectrum includes topographic data (Fig. S5c) and shows interesting periodic features at multiple biases, demonstrated representatively in Fig. S5d-g. At biases well within the band-gap like region of the STS spectra (i.e., Fig. S5e and S5f), the density of states is strongly localized in the near-edge region and exhibits periodic features with the doubled periodicity discussed in the main text. This periodicity is shown in plots of the line cuts (Fig. S5h). Spectra averaged at periodic sites within the grid are shown in Fig. S5i, displaced vertically for clarity.
These results confirm the emergence of a SiNS edge state, but do not address the effect of the Ag(111) substrate in close proximity to the edge. The Ag(111) surface hosts a surface state that is expected to dramatically influence the spectral signature of the edge state. We observe SiNS pZZ edges adjacent to Ag(111) terraces, as shown in fig. S6a. STS point spectra were acquired on this edge. Line spectra plotted in Fig. S6b demonstrate the absence of periodic modulations along the edge, although the edge does host a strong localization of dI/dV intensity in the perpendicular line spectra plot (Fig. S6c, yellow line denotes topographic boundary). The grid topography (Fig. S6e) and dI/dV maps (S6f-k), along with extracted point spectra (fig. S6d), confirm that the enhanced density of states resides at the edge, but lacks the periodicity observed in Fig. 4 and Fig. S5.
Evidently, the additional charge donated by the metallic Ag(111) surface suppresses the formation of periodic electronic states.
Grid spectra were also acquired along both monolayer and bilayer pAC edges, as shown in Fig. S7.
Fig. S7a shows the topography of a monolayer pAC edge reproduced from fig. 5a. The STS line spectra plot (Fig. S7b) shows localization of the edge state at the topographic edge boundary (yellow line). The grid topography is shown in Fig. S7c. The accompanying dI/dV intensity map (Fig. S7d) shows strongly periodic features in the empty states, which are demonstrated in the site- averaged spectra (Fig. S7e). Although there is some enhancement in the filled states at the edge, the enhancement in empty states is more apparent. The topographic distortions along the edge are indicative of edge reconstruction in this monolayer limit, which are consistent with the periodicity of the enhanced empty states signal. However, the relative rarity of these monolayer pAC edges suggests that they are energetically unfavorable. On the other hand, we observe the direct growth of
bilayer pAC edges (Fig. S7f) commonly and without this evident edge reconstruction. In these, the edge state is shown to be localized at the topographic edge, as shown through STS line plots (Fig.
S7g) and spatial maps of the grid spectra (Fig. S7h-k). These dI/dV maps show periodic features, which are emphasized in the site-averaged point spectra (Fig. S7l).
Supplementary Figures
Figure S1. Atomic structure model for the HCT structure.
Figure S2. Comparison of ab-initio (left panel) and tight binding bands (right panel) for the HCT model structure with two Si(111) bilayers as the substrate. The relative sizes of the blue dots (left panel) indicate the contribution of Ag and the upper layer Si atoms to the bands. The Hamiltonian corresponding to the tight-binding bands underlies the Green’s function used for STM/STS
calculations.
Figure S3. Calculated STM images of the periodic HCT reconstruction at (a) empty states at 0.2 eV and (b) filled states at -0.7 eV, and of the IET reconstruction for (c) empty states at 0.2 eV and (d) for filled states at -0.7 eV.
Figure S4. Atomic structure models and the computed STM images shown in Figs. 4 (a)-(b) and 5 (a)-(b) are shown rearranged and augmented with additional STM linescans through the blue and red shaded areas: (a) pZZ structure and (b) pAC structure.
Figure S5. STS Grid Spectroscopy on ML pseudo-zigzag edge. (a) STM topograph of SiNS monolayer-height pZZ edge (Vsample = -0.6 V, It = 1.0 nA). (b) STS dI/dV spectra obtained across the edge in (a). Yellow line denotes the topographic edge location. (c-g) STS grid topography (c) and dI/dV intensity maps (d-g) at indicated biases. (h) Plot of topographic and dI/dV line cuts along edge, extracted from the grid spectra. (i) Averaged site spectra (offset vertically) from the points indicated in (c).
Figure S6. STS Grid Spectroscopy on pseudo-zigzag edge bordering the SiNS/Ag interface. (a) STM topography image of pZZ edge bordering an Ag(111) terrace (Vsample = 0.3 V, It = 800 pA).
(b,c) STS line spectra plots parallel (b) and perpendicular (c) to the edge demonstrate a constant enhancement of the density of states, localized to the topographic edge (denoted by yellow line in c). (d) Isolated point spectra from regions shown in (e), which demonstrate metallic characteristics (inset) relative to the typically reduced dI/dV measured within the band-gap like region between approximately -500 mV to 800 mV. (e-k) STS topography (e) and dI/dV intensity maps (f-k) at the indicated biases.
Figure S7. STS Grid Spectroscopy on ML and BL pseudo-armchair edges. (a) STM topography of monolayer pAC edge (Vsample = -0.8 V, It = 2.0 nA). (b) STS line spectra plot perpendicular to the edge, demonstrating localization of the edge states to the topographic edge boundary (yellow line).
(c, d, e) grid spectra topography (c) and dI/dV intensity (d), along with the site-averaged point spectra (e). (f) STM topography image of a bilayer step pAC edge (Vsample = -2.0 V, It = 1.0 nA).
(g) STS line spectra plot perpendicular to the edge in (f), demonstrating localization of the edge states to the topographic edge boundary (yellow line). (h-k) Grid spectra topography (h) and dI/dV intensity (i-k) at the indicated biases. (l) Site-averaged point spectra at the regions color-coded to (h).
Figure S8. Spin-polarized calculations for pseudo-zigzag edges. (a) Calculated dispersion and (b) spin-polarization for the pZZ edge structure. Insets below show expanded view with features at the
point. (c-g) Comparison of atomic structure model (c) and images of LDOS (d) and spin polarization resolved along the x, y, and z directions (e-g), respectively.
Figure S9. Spin-polarized calculations for pseudo-armchair edges. (a) Calculated dispersion and (b) spin-polarization for the pAC edge structure. Insets below show expanded view with features at the
point. (c-g) Comparison of atomic structure model (c) and images of LDOS (d) and spin polarization resolved along the x, y, and z directions (e-g), respectively.
Supplementary References
[1] P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
[2] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
[3] G. Kresse and J. Hafner, Phys. Rev. B 48, 13115 (1993).
[4] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
[5] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
[6] H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976).
[7] G.D: Mahan, Many-Particle Physics, 3rd Ed., Kluwer Academics/Plenum Publishers, New York (2000).
[8] A.L. Fetter and J.D. Walecka, Quantum Theory of Many-Particle Systems. Dover (2003).
[9] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005).
[10] T.H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, and L. Fu, Nature Comm., 3, 982 (2012).
[11] T. N. Todorov, G. A. D. Briggs, and A. P. Sutton, J. Phys.: Condens. Matter 5, 2389 (1993).
[12] J. B. Pendry, A. B. Pretre, and B. C. H. Krutzen, J. Phys.: Condens. Matter 3, 4313 (1991).
[13] J. A. Nieminen, E. Niemi, and K.-H. Rieder, Surf. Sci. 552, L47 (2004).
[14] J. Nieminen, H. Lin, R. S. Markiewicz, and A. Bansil, Phys. Rev. Lett. 102, 037001 (2009).
[15] M. Ezawa and N. Nagaosa, Phys. Rev. B 88, 121401(R) (2013).
